NCERT Solutions
Class 11 Maths
Mathematical Reasoning
Ex.14.5 Q.1
Show that the statement p: “If x is a real number such that x3 + 4x = 0, then x is 0” is true by
(1) direct method
(2) method of contradiction
(3) method of contra positive
p: “If x is a real number such that x3 + 4x = 0, then x is 0”.
Let q: x is a real number such that x3 + 4x = 0
r: x is 0.
(1) To show that statement p is true, we assume that q is true and then show that r is true.
Therefore, let statement q be true.
So, x3 + 4x = 0
=> x (x2 + 4) = 0
=> x = 0 or x2+ 4 = 0
However, since x is real, it is 0.
Thus, statement r is true.
Therefore, the given statement is true.
(2) To show statement p to be true by contradiction, we assume that p is not true.
Let x be a real number such that x3 + 4x = 0 and let x is not 0.
Therefore, x3 + 4x = 0
=> x (x2 + 4) = 0
=> x = 0 or x2 + 4 = 0
=> x = 0 or x2 = – 4
However, x is real. Therefore, x = 0, which is a contradiction since we have assumed that x is not 0.
Thus, the given statement p is true.
(3) To prove statement p to be true by contra positive method, we assume that r is false and prove that q must be false.
Here, r is false implies that it is required to consider the negation of statement r.
This obtains the following statement:
∼r: x is not 0.
It can be seen that (x2 + 4) will always be positive.
x ≠ 0 implies that the product of any positive real number with x is not zero.
Let us consider the product of x with (x2 + 4).
=> x (x2 + 4) ≠ 0
=> x3 + 4x ≠ 0
This shows that statement q is not true.
Thus, it has been proved that
∼r ⇒∼q
Therefore, the given statement p is true.